Question: Emily is 2 times as old as William. 30 years ago, Emily was 7 times as old as William. How old is Emily now?
Solution: We can use the given information to write down two equations that describe the ages of Emily and William. Let Emily's current age be $e$ and William's current age be $w$ The information in the first sentence can be expressed in the following equation: $e = 2w$ 30 years ago, Emily was $e - 30$ years old, and William was $w - 30$ years old. The information in the second sentence can be expressed in the following equation: $e - 30 = 7(w - 30)$ Now we have two independent equations, and we can solve for our two unknowns. Because we are looking for $e$ , it might be easiest to solve our first equation for $w$ and substitute it into our second equation. Solving our first equation for $w$ , we get: $w = e / 2$ . Substituting this into our second equation, we get: $e - 30 = 7($ $(e / 2)$ $- 30)$ which combines the information about $e$ from both of our original equations. Simplifying the right side of this equation, we get: $e - 30 = \dfrac{7}{2} e - 210$ Solving for $e$ , we get: $\dfrac{5}{2} e = 180$ $e = \dfrac{2}{5} \cdot 180 = 72$.